Mahdieh Zaker

Mahdieh Zaker, Amy Nejati, Abolfazl Lavaei

Infinite networks are complex interconnected systems comprising a countably infinite number of subsystems, for which no fixed upper bound on the number of participating subsystems is specified a priori since it may vary over time as agents join or leave (e.g., vehicles in traffic). In such scenarios, the presence of infinitely many subsystems within the network renders the existing analysis frameworks tailored for finite networks inapplicable to infinite ones. This paper is concerned with offering a data-driven approach, within a compositional framework, for the safety certification of infinite networks with both unknown mathematical models and unknown interconnection topologies. Given the immense computational complexity stemming from the extensive dimension of infinite networks, our approach capitalizes on the joint dissipativity-type properties of subsystems, characterized by storage certificates. We introduce innovative compositional data-driven conditions to construct a barrier certificate for the infinite network leveraging storage certificates of its unknown subsystems derived from data, while offering correctness guarantees for network safety. We demonstrate that our compositional data-driven reasoning eliminates the requirement for checking the traditional dissipativity condition, which typically mandates precise knowledge of the interconnection topology. We illustrate our data-driven results on two physical infinite networks with unknown models and interconnection topologies.

Mahdieh Zaker, David Angeli, Abolfazl Lavaei

This work focuses on a compositional data-driven approach to verify incremental global asymptotic stability (delta-GAS) over interconnected homogeneous networks of degree one with unknown mathematical dynamics. Our proposed approach leverages the concept of incremental input-to-state stability (delta-ISS) of subsystems, characterized by delta-ISS Lyapunov functions. To implement our data-driven scheme, we initially reframe the delta-ISS Lyapunov conditions as a robust optimization program (ROP). Due to the presence of unknown subsystem dynamics in the ROP constraints, we develop a scenario optimization program (SOP) by gathering data from trajectories of each unknown subsystem. However, since the measured one-step transition data are corrupted by noise with a known bound on its norm, rendering the proposed SOP intractable, we introduce an auxiliary SOP that explicitly accommodates noisy measurements. We solve the auxiliary SOP and construct a delta-ISS Lyapunov function for each subsystem with unknown dynamics. We then leverage a small-gain compositional condition to facilitate the construction of an incremental Lyapunov function for an unknown interconnected network based on the data-driven delta-ISS Lyapunov functions of its individual subsystems, while providing correctness guarantees, incorporating the bound on the noise norm. We demonstrate that our data-driven compositional approach reduces the sample complexity to the subsystem level. To validate the effectiveness of our approach, we apply it to an unknown controlled physical nonlinear homogeneous network of degree one, comprising 10000 subsystems. By gathering noisy data from each unknown subsystem, we demonstrate that the interconnected network is delta-GAS with a correctness guarantee.

Mahdieh Zaker, Omid Akbarzadeh, Behrad Samari et al.

In this work, we propose a compositional scheme based on small-gain reasoning to synthesize safety controllers for interconnected stochastic hybrid systems. In our proposed setting, we first offer an augmented scheme that characterizes each stochastic hybrid subsystem, endowed with both continuous evolution and instantaneous jumps, within a unified framework including both scenarios, implying that its state trajectories coincide with those of the original hybrid subsystem. We then introduce the concept of augmented control sub-barrier certificates (A-CSBCs) for each subsystem, thereby enabling the construction of an augmented control barrier certificate (A-CBC) for an interconnected network (from A-CSBCs of its subsystems) along with its safety controller under small-gain compositional conditions. We eventually leverage the constructed A-CBC to derive a guaranteed lower bound on the safety probability of the interconnected network. While in a monolithic scheme the computational complexity of synthesizing a control barrier certificate via sum-of-squares (SOS) optimization scales polynomially with the overall network size, the proposed compositional framework reduces this dependence to the subsystem size. We illustrate the efficacy of the proposed approach on an interconnected network comprising 1000 stochastic hybrid subsystems with nonlinear dynamics under two distinct interconnection topologies.

Mahdieh Zaker, Andrii Mironchenko, Amy Nejati et al.

This paper develops a direct data-driven framework for infinite networks with unknown nonlinear polynomial subsystems, enabling the synthesis of controllers that ensure the entire network is uniformly globally asymptotically stable (UGAS). To address scalability challenges arising from high dimensionality, we develop a data-driven approach to construct an input-to-state stable (ISS) Lyapunov function and its corresponding controller for each unknown subsystem using only a single set of noise-corrupted input-state trajectories collected from that subsystem. Once each subsystem admits a data-driven ISS Lyapunov function, we leverage a compositional small-gain framework for infinite-dimensional spaces to construct a global control Lyapunov function and its associated controller, thereby ensuring UGAS of the entire infinite network. The effectiveness of the proposed data-driven approach is demonstrated through three case studies, including infinite networks of spacecraft, Lorenz chaotic systems, and an academic example with a state-dependent control input matrix.